Physical Sciences and Mathematics Commons^™

Prescribing The Q¯ ′ -Curvature On Pseudo-Einstein Cr 3-Manifolds, Ali Maalaoui

Mathematics

In this paper we study the problem of prescribing the Q ¯ ′ -curvature on embeddable pseudo-Einstein CR 3-manifolds. In the first stage we study the problem in the compact setting and we show that under natural assumptions, one can prescribe any positive (resp. negative) CR pluriharmonic function, if ∫ M Q ′ d v θ > 0 (resp. ∫ M Q ′ d v θ < 0 ). In the second stage, we study the problem in the non-compact setting of the Heisenberg group. Under mild assumptions on the prescribed function, we prove existence of a one parameter family of solutions. In fact, we show that one can find two kinds of solutions: normal ones that satisfy an isoperimetric inequality and non-normal ones that have a biharmonic leading term.

The available download on this page is the author manuscript accepted for publication. This version has undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading …

Some Results On A Class Of Functional Optimization Problems, David Rushing Dewhurst

Graduate College Dissertations and Theses

We first describe a general class of optimization problems that describe many natu- ral, economic, and statistical phenomena. After noting the existence of a conserved quantity in a transformed coordinate system, we outline several instances of these problems in statistical physics, facility allocation, and machine learning. A dynamic description and statement of a partial inverse problem follow. When attempting to optimize the state of a system governed by the generalized equipartitioning princi- ple, it is vital to understand the nature of the governing probability distribution. We show that optimiziation for the incorrect probability distribution can have catas- trophic results, e.g., …

Quantum Groups And Knot Invariants, Greg A. Hamilton

Honors Theses

Knot theory arguably holds claim to the title of the mathematical discipline with the most unusually diverse applications. A knot can be defined topologically as an embedding of S1 in R3. Naturally, two knots are topologically equivalent if one cannot be smoothly deformed into the other. The question of whether two knots are equivalent is highly non-trivial, and so the question of knot invariants used to distinguish knots has occupied knot theorists for over a century. Knot theory has found application in statistical mechanics [1], symbolic logic and set theory [2], quantum fi theory [3], quantum computing [4], etc. …

Langer's Method For The Calculation Of Escape Rates And Its Application To Systems Of Ferromagnets, Gerard Duff

Doctoral

This work is a study of the application of a theory proposed by J. S. Langer (J.S. Langer, Statistical Theory of the Decay of Metastable States, Annals of Physics 54, 258-275 (1969)) for the calculation of the decay rate (relaxation rate) of a metastable state. The theory is set in the context of statistical mechanics, where the dynamics of a system with a large number of degrees of freedom (order 1023) are reduced to N degrees of freedom, where N is small, when a steady state or equilibrium position is maintained by the entire system. In this thesis N equals …

Statistical Mechanics Of Farey Fraction Spin Chain Models, Jan Fiala

Electronic Theses and Dissertations

This thesis considers several statistical models defined on the Farey fractions. Two of these models, considered first, may be regarded as "spin chains", with long-range interactions, another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle-Perron-F'robenius operator), which is defined using the maps (presentation functions) generating the Farey "tree". The spectrum of this operator was completely determined by Prellberg. It follows that all these models have a second-order phase transition with a specific …

Objectivity, Information, And Maxwell's Demon, Steven Weinstein

Dartmouth Scholarship

This paper examines some common measures of complexity, structure, and information, with an eye toward understanding the extent to which complexity or information‐content may be regarded as objective properties of individual objects. A form of contextual objectivity is proposed which renders the measures objective, and which largely resolves the puzzle of Maxwell's Demon.

Limit Theorems In The Area Of Large Deviations For Some Dependent Random Variables, Narasinga Rao Chaganty, Jayaram Sethuraman

Mathematics & Statistics Faculty Publications

A magnetic body can be considered to consist of n sites, where n is large. The magnetic spins at these n sites, whose sum is the total magnetization present in the body, can be modelled by a triangular array of random variables (X(n) 1,..., X(n) n). Standard theory of physics would dictate that the joint distribution of the spins can be modelled by dQ_n(x) = z_n^-1 exp[ -H_n(x)]Π dP(x_j), where x = (x₁,..., x_n) ∈ Rⁿ, where H_n is the Hamiltonian, z_n is …